cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A341423 Number of positive solutions to (x_1)^2 + (x_2)^2 + (x_3)^2 + (x_4)^2 <= n^2.

Original entry on oeis.org

1, 5, 32, 94, 219, 437, 804, 1362, 2177, 3271, 4768, 6708, 9227, 12381, 16254, 20954, 26707, 33461, 41480, 50884, 61703, 74183, 88606, 104862, 123481, 144241, 167604, 193648, 222799, 254731, 290244, 329512, 372545, 419661, 470822, 526646, 587481, 653505
Offset: 2

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A046895, A055403, A055410, A063730, A224213, A253663, A302995, A341424, A341425, A341426, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 4):
seq(a(n), n=2..39);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^4/(16 (1 - x)), {x, 0, n^2}], {n, 2, 39}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^4 / (16 * (1 - x)).

A341424 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n^2.

Original entry on oeis.org

6, 51, 177, 547, 1348, 2958, 5574, 10084, 16974, 27450, 41970, 62671, 90216, 128082, 175867, 238018, 316373, 414998, 534094, 682144, 859705, 1075165, 1326551, 1627896, 1976582, 2390057, 2862607, 3411273, 4039483, 4760419, 5571729, 6500650, 7541560, 8722096
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055404, A055411, A175360, A253663, A302995, A340481, A341400, A341423, A341425, A341426, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 5):
seq(a(n), n=3..36);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^5/(32 (1 - x)), {x, 0, n^2}], {n, 3, 36}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^5 / (32 * (1 - x)).

A341425 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n^2.

Original entry on oeis.org

7, 48, 331, 1269, 3698, 9382, 20927, 42683, 79844, 142173, 238810, 387615, 603589, 915324, 1345294, 1939221, 2729723, 3783313, 5138567, 6895632, 9108626, 11909496, 15362753, 19642539, 24832744, 31179476, 38757032, 47877886, 58647957, 71447776, 86391220
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055405, A055412, A175361, A253663, A302995, A340905, A341401, A341423, A341424, A341426, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 6):
seq(a(n), n=3..33);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^6/(64 (1 - x)), {x, 0, n^2}], {n, 3, 33}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^6 / (64 * (1 - x)).

A341426 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n^2.

Original entry on oeis.org

1, 71, 491, 2522, 9263, 27723, 71480, 163908, 345657, 679802, 1252185, 2203724, 3715206, 6041979, 9510283, 14591324, 21788606, 31894205, 45741815, 64467383, 89363919, 122254946, 164721244, 219526449, 289133792, 377013829, 486522424, 622759365
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055406, A055413, A253663, A302995, A340906, A341396, A341402, A341423, A341424, A341425, A341427, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 7):
seq(a(n), n=3..30);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^7/(128 (1 - x)), {x, 0, n^2}], {n, 3, 30}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^7 / (128 * (1 - x)).

A341427 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_8)^2 <= n^2.

Original entry on oeis.org

1, 45, 767, 4452, 21178, 74452, 224313, 586035, 1387583, 2999430, 6102276, 11656386, 21282969, 37159993, 62687904, 102213426, 162345824, 251064745, 379922217, 562833191, 819351646, 1171991382, 1651937498, 2294227971, 3147090871, 4263499419
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055407, A055414, A253663, A302995, A340915, A341397, A341403, A341423, A341424, A341425, A341426, A341428, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 8):
seq(a(n), n=3..28);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^8/(256 (1 - x)), {x, 0, n^2}], {n, 3, 28}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^8 / (256 * (1 - x)).

A341428 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n^2.

Original entry on oeis.org

1, 46, 760, 7751, 43910, 186098, 652710, 1943742, 5178030, 12411211, 27773308, 57798904, 114152429, 214399664, 387571706, 673189698, 1135916808, 1857320784, 2966816950, 4623984661, 7066527283, 10577150039, 15589368584, 22580091614, 32256768126
Offset: 3

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055408, A055415, A253663, A302995, A340946, A341398, A341404, A341423, A341424, A341425, A341426, A341427, A341429.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 9):
seq(a(n), n=3..27);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^9/(512 (1 - x)), {x, 0, n^2}], {n, 3, 27}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^9 / (512 * (1 - x)).

A341429 Number of positive solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n^2.

Original entry on oeis.org

56, 1108, 12098, 84624, 439371, 1785368, 6078048, 18139393, 48586117, 118929400, 270250734, 578320470, 1169522013, 2261784392, 4193751331, 7509793133, 13008356489, 21921125415, 35951569269, 57666975238, 90464266824, 139295784464, 210514511189, 313228848537
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 11 2021

Keywords

Links

Crossrefs

Cf. A000122, A001182, A055409, A055416, A253663, A302995, A340947, A341399, A341405, A341423, A341424, A341425, A341426, A341427, A341428.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0, 1, `if`(n=0, 0,
      add((s->`if`(s>n, 0, b(n-s, k-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n^2, 10):
seq(a(n), n=4..27);  # Alois P. Heinz, Feb 11 2021
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x] - 1)^10/(1024 (1 - x)), {x, 0, n^2}], {n, 4, 27}]

Formula

a(n) is the coefficient of x^(n^2) in expansion of (theta_3(x) - 1)^10 / (1024 * (1 - x)).

A303485 a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=1} x^(k^3))^n.

Original entry on oeis.org

1, 1, 1, 8, 66, 512, 5269, 57459, 711742, 9610222, 139735699, 2183555015, 36543300668, 649320343729, 12174674648730, 240360451018461, 4975239937954534, 107600744797471150, 2426579187889852885, 56901290353169050995, 1384258146777832889697
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Comments

Number of positive solutions to (x_1)^3 + (x_2)^3 + ... + (x_n)^3 <= n^3.

Links

Crossrefs

Cf. A000578, A010057, A051344, A280618, A298672, A302995, A303169.

Programs

  • Mathematica
    Join[{1}, Table[SeriesCoefficient[1/(1 - x) Sum[x^k^3, {k, 1, n}]^n, {x, 0, n^3}], {n, 20}]]
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.